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A correlated basis-function description of 16O with realistic interactions
NUCLEAR PHYSICS A
Nuclear Physics A567 (1994) 329-340 North-Holland
A correlated basis-function description of 160 with realistic interactions * M.C. Bosch ~e~a~tarne~t~de Fisica ~~derna, Univer~idadde Granada, Granada, E-18071 Spain Received 27 February 1992 (Revised 30 March 1933) Abstract The correlated basis-function theory is applied at the lowest order to analyze the ground state and low-energy spectrum of the 160 nucleus. Results are quoted for both the Urbana and the Argonne 0~4 nucleon-nucleon interactions, The work includes state-dependent correlations and their radial components are determined by solving a set of Euler-Lagrange equations. The matrix elements are computed by using a cluster expansion and the sequential condition is imposed in order to insure convergence. The results clearly disagree with the experimentai values.
1. Introduction In this work we apply the lowest order of the correlated basis-function (CBF) theory [l-3] to describe the ground state and the low-energy spectrum of the I60 nucleus. Our aim is to extend the calculation quoted in refs. [4,5], hereafter to be denoted by I and II, respectively. Therein the 06 and 0s potentials were used to study the ground states and spectra of light nuclei. In this paper we will show how the extension to o i4 potentials can be made. A systematic description of the theory used and the technical details of its application can be found in I and II. The nucleus is considered as a set of non-relativistic nucleons interacting via a two-body potential with 14 operatorial components of the form 0(1,2) = 2
v;(r)@‘),
(11
i=l
{~“‘)&
=
{l,Ql
.%~I
.r2,
(a1
~az)(zl.22),s12,s12
(21 .72),
L.S,L.S(7,.Z2),L2,L2(d1.Q2),L2(2,.Z2), L2(al.62)(Z,‘~2),(L’S)2,
tL-s)2(vz2)}.
(2)
The radial parts are taken from the Urbana [ 61 or, alternatively, Argonne [ 73 014nucleonnucleon potentials. * Work supported by DGICYT (under contract PB90-0873) and funta de ~ndal~cia. 03759474/94/$07.00 0 1994 - Eisevier Science B.V. All rights reserved SSDI 0375-9474(93)E0300-W
330
MC. Bmxi / Realistic ~n~erac~i~ns 2. Formalism
In the CBF theory the physical A-particle correlated
states are written as linear combinations
of
states 1t,um), Y/(p) = p$‘Iv/nt), m
which are obtained orthogonal
by applying
shell-model
We have considered
a correlation
(3)
operator
F onto a set of normalized
and
states lQfm), i.e.,
uncorrelated
the operator F to have the general structure
F =
S{n[Fz(i,_i)},
(5)
i
operator
Fz(i, j)
and F2 is the two-body correlation,
= F’(i,
j)P+
+ F-(i,
which is taken
j)P-,
(6)
with Fk = _&r)Pt
+ _&Y)QP~
+ .G%)U
-
(k
QV’3
=
(7)
+,-I,
Pk (k = +, - 1, P, and 9 being projectors on even and odd orbital angular-momentum states, and on singlet and triplet spin states, respectively, and
Q =
$12
+
(8)
9’3,
where Siz is the normal tensor operator. The shell-model uncorrelated states are antisymmetric
functions with well-defined
total
angular momentum and isospin, built up with single-particle wave functions taken as ha~onic-oscillator states with the constant izo as the only free parameter. Our basis is composed of, in addition to the ground state, where all the OsOp orbitals are occupied, a set of one-particle-one-hole and two-particle-two-hole states where one or two nucleons have been promoted from the p-shell to the next sd shells, respectively, in all possible forms. The energy eigenvalues and the amplitudes C$’ in Eq. (3) are obtained by solving the generalized
eigenvalue
problem ~t(~ml~l~n) m
- E(~ml~n)K’m
= 0.
(9)
Due to the presence ofthe correlation factor F, it is necessary to compute matrix elements of A-body non-separable operators. To do this we use the CIY cluster expansion, as referred to in I, which is based upon a modified version of the Iwamoto-Yamada (IY)
A4.C. Boscri /Realistic
expansion
interactions
331
[8,9]. Up to the second order, the A-body matrix elements
are given by the
following equations: ]H]w ) = A(+ I&d H mm = (w mm m2m
(~mlHIvnUn)=
0
;
A ) + 2 m 0
(4mlK;klbz)
+ ;hn
wnl Vzwd
3
Wmm + f&n),
(10)
(11)
where Nmn = (&?l$n) =
;
~9mlhvL2) - 1IA),
(12)
0
G-c = ;[wJ),
[T(l) + 7-(2),fi(1,2)11 +
(13)
F2(1,2)0(1,2)F2(1,2).
All matrix elements appearing in these equations are calculated in terms of two-body matrix elements by means of extensive fractional parentage expansions and Racah algebra. The uncorrelated basis states as well as the CIY cluster expansion used were fully described in I, where a detailed graphical notation and some examples of two-body matrix elements can be found. The main difference with respect to the previous 06 and vs calculations is that it is no longer possible to reduce the corresponding two-body effective interaction, Vdiffy, to a v 14form. With the correlation considered, see Eq. (7)) the effective interaction
Vi; takes the general form [ 3 1: vg$ =
[v,q:; + V-T; L2] P,Pk
c
(T,k)=(l,+),(O,-) +
c
[ v;,:f$
+
f+,::
s12
+
v-;&L
.s
(T,k)=(O,+),(l,-)
(14) where (15) (16)
(17)
M.C. Bomi / Realistic interactions
332 ylTk T,eff
1 +[-+p+fvp+fvL,T -p-y _vi:2 __!!$ 1(j$)2 +[-pg+gf-y +;vL’srz +4;;Ifi”f3” +$vp)‘g-$vp)2, (18) 1 h2 [ 1 -2v~T-3v;&2~~ +[&!; 1j$j& 2
=
iv;’
1Tk
VLS,eff
+
p;T-
p;fs’
iVL]T+
=
+
p&
p--T
+
+
+v;fsi
g-$
-
;a;
(hk)’
(f2k)2
[
+
$tiT
+ fVt’s’2 + --;T
1
(hkJ2
(19)
T/lTk L,eff
=
[itiT
-
g-&]
(hk)2
ITk
VLS2,eff VK&T ITk
VTL,eff
= =
[itiT [iVLIT
+ yL&] +
pLl&]
=
+
[pL!T
+
p-i&]
(f3k)2,
tiixhk)2,
(21)
Lf2k)2 + [-fV;l’(f2k)2
+
(20)
[-ivL”
&g] -
(hk12,
p&q
(_&k)2,
(22) (23)
and the I/KsTstands for the various components of the bare potential. The next step to compute the energy is to calculate the set of six correlation functions {fik} introduced in Eq. (7). They are obtained by using the lowest-order constrained variational method, LOCV (see I and references therein). Briefly, to do this we first obtain the expression of the expectation value of the two-body operator V$K, calculated on the ground state, which results in a combination of two-body matrix elements. After transforming this to the center-of-mass and relative coordinates the center of mass, we find the following result: rrG,%
+
lo+ [VLSZ,eff
+ e,:a3LJ’:
+ r9c,:;
and integrating
it over
+ vKqo,r;.lPlfl
3P+ Ls2 + 9V;&,p5s2]
(24)
where py = (P/27r)3’2exp(-pr2/2)
[ +‘-
jlr4 + ~fi’r”],
(25)
pC = (B/2a)312exp(-J3r2/2)4pr4,
(26)
p4 = (P/27r)3/2exp(-j3r2/2)2/32r6,
(27)
M.C. Bosui /Realistic interactions
333
PL- = (~/2n)312exp(-~r2/2)8~r4,
(28) (29)
pts2= (j?/2rr)3/2exp(-~r2/2)$$r4.
(30)
Due to the closed-shell structure of the 160 nucleus, the other ~mponents of the effective potential in Eq. (14) give zero contribution. After imposing the conditions P(O)
= 0,
(311
.#?w
= 1,
(32)
0,
(33)
a dr
,=d
=
where the parameter d is the healing distance, Eq. (24) allows us to obtain the EulerLagrange equations for the two-body correlation operator: id2fk +
dr2 (34) in the singlet channel, with (k, T) = (+, 1) and (-,O),and
~~~~+~(~)2(~)2
dZfZk ---
’
dr2
Ii2P,f [
(
+P,“w!’
- pa
EL p;
= $
vc?'+ 2vlT - lq$ + 2v;r + p~2(~v~)
+ 3v,‘,r, + 2:;
I>
+ $ >
fi
V:,&2V;T-3V-&2$+
J;“,
(35)
d2ffk dr2
---in 1 7i2 pE
+pkLwY
=2
h2 (
V$’ - 4ViT - 2V;; + 4ViT + 6V;& + 4;;
p: [
(
>
I)
+ vz!L!d
+ ,$
s:
2V-j--4V;T-6V:s:-4;f )
r,“,
(36)
in the triplet channel, with (k, 7’) = ( + , 0 ) and ( - ,1) ,,lf being the Lagrange multipliers. In these last expressions we have obtained an hermitian form for the differential equations
M.C. Bomi /Realistic interactions
334
TABLE 1 Energy per nucleon calculations, in MeV, for Urbana and Argonne 0~4interactions. The center-of-mass correction has been considered 014
Eo
E2p211
d(fm)
/iw(MeV)
Urbana Argonne
-4.58 -4.35
-5.32 -5.08
2.17 2.28
23 22
by using the set of modified
correlation
functions,
Jk (~1 = C&(r) )112J;k (rf .
(37)
At this stage the numerical calculation can be performed, as described in I. In this case we have imposed the sequential condition (again see I and references therein)
t3(ijlFi2’(1,2)
- ljij-ji)
= 0,
(38)
in order to have control over the cluster expansion, which we have truncated at the second order. Note that this condition allows us to select the particular value of the healing distance d which satisfies the constraint for each value of the variational parameter hw and, in this way, we can look for the minimum of the energy. The energies obtained at this first level of the calculation will be referred to as Ea. Once all the variational parameters and correlation functions have been determined, we correct the result for the ground-state energy by allowing the mixing of the ~0 state with the V2p2hstates. The energies calculated in this way will be denoted as Ezp2h. Finally, we obtain the excitation spectra by considering the whole basis allowed in each case: vlpih states for negative-parity
excited states and ~0 and v&,zh states for positive parity.
3. Results and final comments Fig. 1 shows the energy per nucleon EO as a function of the parameter fro for both the Urbana (solid curve) and the Argonne (dashed curve) ~14 interactions. In Table 1 we show, for each interaction, the minima of EO and E 2$h together with the co~esponding values of the fiw and d parameters. The results do not include Coulomb terms, and the center-of-mass correction is taken into account by simply substracting $ho, since we have chosen the single-particle wave functions as harmonic-oscillator states. As is apparent, our results are considerably far from the experimental value (- 7.07 MeV/nucleon). The reason for this discrepancy is not fully understood, however some points deserve commenting upon. Firstly, we have not estimated the importance of many-body cluster terms. Though we have tried to minimize their effects through the sequential condition, Eq. (38), sundry calculations have shown that their consideration increases the binding energy, and the
335
MC. Romi 1 Realistic interactions -1.0
-2.0
F z i
-3.0
W
-4.0
-5.o_ 5.
10.
15.
20.
25.
pie
[MeV]
30.
35.
40.
Fig. 1. The Eo energy per nucleon as a function of the variational parameter fro; the center-of-mass correction has been considered; full (dashed) line corresponds to Urbana (Argonne) v14interaction.
same occurs if a three-nucleon potential is added to the two-nucleon one [10-l 31. As an example, here we mention a recent work of Pieper et al. [ 141 in which the values E/A = -5.8 MeV for I60 and E/A = -5.6 MeV for 4He have been found with the Argonne 014 interaction. In this calculation the central correlation and exchanges are treated to all orders by Monte Carlo integration state-dependent of the Urbana
correlations,
VII three-nucleon
by 1.2 MeV/nucleon
and a cluster expansion
which include an L-S term. Furthermore, potential
[ 15 J is added, the binding
for I60 and 1.4 MeV/nucleon
for 4He. Further
is used for the
if the contribution energy increases calculations
with
more accurate forms of the wave function show that three-body correlations produce a signi~cant increase in the binding energy [ 161. Clearly, any calculation based on clusters expansions must guarantee the convergence of the cluster series. Secondly, our two-body factor correlation in Eq. (7) should have a full 014 structure to be consistent with the interaction, or, at least, L 1S pieces, which will produce new kinds of terms in the effective potential. The absence of these components could be one of the reasons for the lack of binding energy shown by the 014 interactions with respect to our previous calculations with 0s interactions. Although we have not tested how these new terms affect the energy, we expect the effects to be small, but not negligible. The Ezp2h results in Table 1 show, as for the 08 interaction, that the correct construction of the nuclear wave function plays an important role. In this respect the admixtures of
M.C. Boscci /Realistic interactions
336
TABLE 2 Kinetic energy and cont~butions in MeV/nucieon of each component of the ut4 interaction to the Eo value of the energy
014
hw(MeV)
Urbana Argonne
23 22
Contributions
T 14.18 18.51
VC
-35.72 -24.08
VLS
VT
-8.77 -23.91
-0.10 0.56
VL
1.21 2.99
VLS2
-0.18 -2.23
cv
-43.56 -46.47
TABLE 3 in MeV/nucleon of each component of the effective potential V:& to the Eo value
v14
fiw(MeV)
Urbana Argonne
23 22
-30.66 -29.74
0.26 0.02
1.02 1.66
-29.38 -28.06
-4.58 -4.35
TABLE 4 Same as Table 1 for Iio = 15MeV Eo
E2,,2h
-3.21 -3.16
-3.81 -3.76
014
Urbana Argonne
d(h)
2.05 2.17
states produce a non-negligible effect, the corresponding gain in energy being about 0.7 MeV per nucleon; this fact can be related to the extreme simplicity of the correlation operator used. Tables 2 and 3 show the contribution in MeV/nucleon of each component of the u14 interaction and V$, potential, respectively, to the E. value of the energy. These contributions are related by E. = %co 64
=
+ Ef
c v, , eff
(401
K
where K = {C, T, LS, L, LS2). From these tables we can conclude that, as expected, the central and tensor parts of the interaction give the most important contribution to the total energy, the contributions of the remaining terms being non-negligible.
337
;I= I--
2-_ o-_ 1--
2-4-----
z-4--
2-4--
3-_ 2-
3-_
3-
2-
2--
0-_ ‘2--
o- = 2-
1------
1--
3-----
3--
2p’----
-
20-1-3--
9
F
I-2--
t
1-----
6
3--
~
3
Exp.
I’rb-VI4
ArgV1.l
RX-V8
-
Fig. 2. Low negative-parity states with isospin T=O.
The optimal
values found for frw in the ground-state
minimization
are too large (al-
though fiw does not correspond to the mean-field ha~oni~-oscilIator potential, because of the central-field modification introduced by the Jastrow correlation through the CBF mechanism). This fact may be due to the inadequacy of harmonic-oscillator single-particle states, and a way to improve this deficiency would be to carry out a Hartree-Fock determination of single-particle
wave functions
from the Jastrow generated effective interaction,
which will imply a highly non-trivial double self-consistent calculation, given that the effective interaction depends on the correfation function, which in turn depends on the nuclear density. Few calculations of energies of light nuclei using 014 interactions exist to compare our work with, although most of the theoretical effort has been devoted in recent years [ 1O161 to the aIpha particle. The corresponding variationai results for the ground-state energy are of 5.7 [ 151 and 5.7 [ 151-5.8 [ 131 ~ev~nuc~eon with the Urbana and Argonne at4 interactions, respectively. These values are lowered to 7.3 [ 151 and 7.1 [ 15]-7,O [ 131
M.C. Bosch / Realistic interactions
338
E
I
i
[M&l -
1
T=l
1-_
21 -
;‘4-
21 -
‘L-. 3--_
18 -
,-,2-_
2_
“-
1-
1-1-
2--
7-p
9--
73-
4-‘A3-----~~
1I--
15 -
? :j--
12 -
9--
‘-
o-
I
Fig. 3. Low negative-parity states with isospin T= 1.
MeV/nucleon, respectively, when the three-nucleon Urbana VII potential is added. Our calculations for the alpha particle and other nuclei in the p-shell [ 171 also show a lack of binding energy in comparison with those results previously mentioned, making it clear that our choice of the correlation factor is inadequate. The calculated spectra are shown in Figs. 2-4, together with the experimental
levels
[ 181 and the previous us calculations [ 51. A value of hw = I5 MeV has been used for both interactions considered. The reason for choosing this value instead of our optimal values shown in Table 1 is twofold. Firstly, the levels theoretically determined by using the optimal values are too high in excitation and, secondly, this was the value used in II for the comparison with several other ug calculations. Table 4 shows the corresponding energy values for this choice of fiw. As is apparent from these figures, the new terms in the potential have not been able to improve the results previously obtained for the 0s interaction, at least at the level of the calculation described above. As in the us interaction, the present approach fails completely in the quantitative description of the excitation energies, and a large shift of the whole calculated spectrum is found with respect to the experimental data. The reasons for this shift, which is unlikely to be resolved by adding components to the potential or to the correlation operator, are not clear. It would probably be convenient to consider other effects such as the excitations of the center-of-mass motion and, in addition, to try
339
1+-
1*-
2*I+-
2+---I”= ,g+--...--
4+2f
2+
3+-
:3+-
0+-
0+-
2+-
.>+_
p----0*5
t
1
&-VI
Esp.
1
Fig. 4. Low positive-parity states with isospin T=O.
different formuIations of the CBF scheme. In any case, and before tackling all the consequent work involved, we are interested in testing if the new terms in the potential, at the level of calculation that we have exposed here, have a significant repercussion on the co~espo~di~g excited spectra of the other nuclei in the p-shell with respect to the results previously obtained for the us nucleonnucleon potentials. Work in this direction is in progress. The author is grateful to R. Guardiola for his very valuable guidance, and also wishes to thank A.M. Lallena for his assistance in the drawings. A critical reading of the paper by S. Fantoni is acknowledged.
References il] 1.W. Clark and P. Westhaus,
Phys. Rev.
141 619661 833; 143 tg966) %I
340
M.C. Bomi /Realistic interactions
[2] J.W. Clark in Progress in particle and nuclear physics, vol. 2, ed. D.H. Wilkinson (Pergamon, Oxford, 1979) [3] S. Fantoni, B.L. Friman and V.R. Pandharipande, [4] [5] [ 61 [7] [8] [9]
[lo] [ 111 [ 121 [13] [ 141 [ 151 [16] [ 171 [ 181
Nucl. Phys. A386 (1982) 1; A399 (1983)
t.C. Bosch and R. Guardiola, Nucl. Phys. A476 (1988) 471 R. Guardiola and M.C. Bosch, Nucl. Phys. A489 (1988) 45 I.E. Lagaris and V.R. Pandharipande, Nucl. Phys. A359 (1981) 331 R.B. Wiringa, R.A. Smith and T.L. Ainsworth, Phys. Rev. C29 (1984) 1207 J.W. Clark and P. Westhaus, J. Math. Phys. 9 (1968) 131,149 M.L. Ristig, W.J. Ter Low and J.W. Clark, Phys. Rev. C3 (1971) 1504 J. Carlson and V.R. Pandharipande, Nucl. Phys. A371 (1981) 301 J. Carlson, V.R. Pandharipande and R.B. Wiringa, Nucl. Phys. A401 (1983) 59 R.B. Wiringa, Nucl. Phys. A401 (1983) 86 J. Carlson, Phys. Rev. C38 (1988) 1879 SC. Pieper, R.B. Wiringa and V.R. Pandharipande, Phys. Rev. Lett. 64 (1990) 364 R. Schiavilla, V.R. Pandharipande and R.B. Wiringa, Nucl. Phys. A449 (1986) 219 J. Carlson, Nucl. Phys. A522 (1991) 185~ M.C. Bosch, to be published F. Ajzenberg-Selove, Nucl. Phys. A460 (1986) 1
Nuclear Physics A567 (1994) 329-340 North-Holland
A correlated basis-function description of 160 with realistic interactions * M.C. Bosch ~e~a~tarne~t~de Fisica ~~derna, Univer~idadde Granada, Granada, E-18071 Spain Received 27 February 1992 (Revised 30 March 1933) Abstract The correlated basis-function theory is applied at the lowest order to analyze the ground state and low-energy spectrum of the 160 nucleus. Results are quoted for both the Urbana and the Argonne 0~4 nucleon-nucleon interactions, The work includes state-dependent correlations and their radial components are determined by solving a set of Euler-Lagrange equations. The matrix elements are computed by using a cluster expansion and the sequential condition is imposed in order to insure convergence. The results clearly disagree with the experimentai values.
1. Introduction In this work we apply the lowest order of the correlated basis-function (CBF) theory [l-3] to describe the ground state and the low-energy spectrum of the I60 nucleus. Our aim is to extend the calculation quoted in refs. [4,5], hereafter to be denoted by I and II, respectively. Therein the 06 and 0s potentials were used to study the ground states and spectra of light nuclei. In this paper we will show how the extension to o i4 potentials can be made. A systematic description of the theory used and the technical details of its application can be found in I and II. The nucleus is considered as a set of non-relativistic nucleons interacting via a two-body potential with 14 operatorial components of the form 0(1,2) = 2
v;(r)@‘),
(11
i=l
{~“‘)&
=
{l,Ql
.%~I
.r2,
(a1
~az)(zl.22),s12,s12
(21 .72),
L.S,L.S(7,.Z2),L2,L2(d1.Q2),L2(2,.Z2), L2(al.62)(Z,‘~2),(L’S)2,
tL-s)2(vz2)}.
(2)
The radial parts are taken from the Urbana [ 61 or, alternatively, Argonne [ 73 014nucleonnucleon potentials. * Work supported by DGICYT (under contract PB90-0873) and funta de ~ndal~cia. 03759474/94/$07.00 0 1994 - Eisevier Science B.V. All rights reserved SSDI 0375-9474(93)E0300-W
330
MC. Bmxi / Realistic ~n~erac~i~ns 2. Formalism
In the CBF theory the physical A-particle correlated
states are written as linear combinations
of
states 1t,um), Y/(p) = p$‘Iv/nt), m
which are obtained orthogonal
by applying
shell-model
We have considered
a correlation
(3)
operator
F onto a set of normalized
and
states lQfm), i.e.,
uncorrelated
the operator F to have the general structure
F =
S{n[Fz(i,_i)},
(5)
i
operator
Fz(i, j)
and F2 is the two-body correlation,
= F’(i,
j)P+
+ F-(i,
which is taken
j)P-,
(6)
with Fk = _&r)Pt
+ _&Y)QP~
+ .G%)U
-
(k
QV’3
=
(7)
+,-I,
Pk (k = +, - 1, P, and 9 being projectors on even and odd orbital angular-momentum states, and on singlet and triplet spin states, respectively, and
Q =
$12
+
(8)
9’3,
where Siz is the normal tensor operator. The shell-model uncorrelated states are antisymmetric
functions with well-defined
total
angular momentum and isospin, built up with single-particle wave functions taken as ha~onic-oscillator states with the constant izo as the only free parameter. Our basis is composed of, in addition to the ground state, where all the OsOp orbitals are occupied, a set of one-particle-one-hole and two-particle-two-hole states where one or two nucleons have been promoted from the p-shell to the next sd shells, respectively, in all possible forms. The energy eigenvalues and the amplitudes C$’ in Eq. (3) are obtained by solving the generalized
eigenvalue
problem ~t(~ml~l~n) m
- E(~ml~n)K’m
= 0.
(9)
Due to the presence ofthe correlation factor F, it is necessary to compute matrix elements of A-body non-separable operators. To do this we use the CIY cluster expansion, as referred to in I, which is based upon a modified version of the Iwamoto-Yamada (IY)
A4.C. Boscri /Realistic
expansion
interactions
331
[8,9]. Up to the second order, the A-body matrix elements
are given by the
following equations: ]H]w ) = A(+ I&d H mm = (w mm m2m
(~mlHIvnUn)=
0
;
A ) + 2 m 0
(4mlK;klbz)
+ ;hn
wnl Vzwd
3
Wmm + f&n),
(10)
(11)
where Nmn = (&?l$n) =
;
~9mlhvL2) - 1IA),
(12)
0
G-c = ;[wJ),
[T(l) + 7-(2),fi(1,2)11 +
(13)
F2(1,2)0(1,2)F2(1,2).
All matrix elements appearing in these equations are calculated in terms of two-body matrix elements by means of extensive fractional parentage expansions and Racah algebra. The uncorrelated basis states as well as the CIY cluster expansion used were fully described in I, where a detailed graphical notation and some examples of two-body matrix elements can be found. The main difference with respect to the previous 06 and vs calculations is that it is no longer possible to reduce the corresponding two-body effective interaction, Vdiffy, to a v 14form. With the correlation considered, see Eq. (7)) the effective interaction
Vi; takes the general form [ 3 1: vg$ =
[v,q:; + V-T; L2] P,Pk
c
(T,k)=(l,+),(O,-) +
c
[ v;,:f$
+
f+,::
s12
+
v-;&L
.s
(T,k)=(O,+),(l,-)
(14) where (15) (16)
(17)
M.C. Bomi / Realistic interactions
332 ylTk T,eff
1 +[-+p+fvp+fvL,T -p-y _vi:2 __!!$ 1(j$)2 +[-pg+gf-y +;vL’srz +4;;Ifi”f3” +$vp)‘g-$vp)2, (18) 1 h2 [ 1 -2v~T-3v;&2~~ +[&!; 1j$j& 2
=
iv;’
1Tk
VLS,eff
+
p;T-
p;fs’
iVL]T+
=
+
p&
p--T
+
+
+v;fsi
g-$
-
;a;
(hk)’
(f2k)2
[
+
$tiT
+ fVt’s’2 + --;T
1
(hkJ2
(19)
T/lTk L,eff
=
[itiT
-
g-&]
(hk)2
ITk
VLS2,eff VK&T ITk
VTL,eff
= =
[itiT [iVLIT
+ yL&] +
pLl&]
=
+
[pL!T
+
p-i&]
(f3k)2,
tiixhk)2,
(21)
Lf2k)2 + [-fV;l’(f2k)2
+
(20)
[-ivL”
&g] -
(hk12,
p&q
(_&k)2,
(22) (23)
and the I/KsTstands for the various components of the bare potential. The next step to compute the energy is to calculate the set of six correlation functions {fik} introduced in Eq. (7). They are obtained by using the lowest-order constrained variational method, LOCV (see I and references therein). Briefly, to do this we first obtain the expression of the expectation value of the two-body operator V$K, calculated on the ground state, which results in a combination of two-body matrix elements. After transforming this to the center-of-mass and relative coordinates the center of mass, we find the following result: rrG,%
+
lo+ [VLSZ,eff
+ e,:a3LJ’:
+ r9c,:;
and integrating
it over
+ vKqo,r;.lPlfl
3P+ Ls2 + 9V;&,p5s2]
(24)
where py = (P/27r)3’2exp(-pr2/2)
[ +‘-
jlr4 + ~fi’r”],
(25)
pC = (B/2a)312exp(-J3r2/2)4pr4,
(26)
p4 = (P/27r)3/2exp(-j3r2/2)2/32r6,
(27)
M.C. Bosui /Realistic interactions
333
PL- = (~/2n)312exp(-~r2/2)8~r4,
(28) (29)
pts2= (j?/2rr)3/2exp(-~r2/2)$$r4.
(30)
Due to the closed-shell structure of the 160 nucleus, the other ~mponents of the effective potential in Eq. (14) give zero contribution. After imposing the conditions P(O)
= 0,
(311
.#?w
= 1,
(32)
0,
(33)
a dr
,=d
=
where the parameter d is the healing distance, Eq. (24) allows us to obtain the EulerLagrange equations for the two-body correlation operator: id2fk +
dr2 (34) in the singlet channel, with (k, T) = (+, 1) and (-,O),and
~~~~+~(~)2(~)2
dZfZk ---
’
dr2
Ii2P,f [
(
+P,“w!’
- pa
EL p;
= $
vc?'+ 2vlT - lq$ + 2v;r + p~2(~v~)
+ 3v,‘,r, + 2:;
I>
+ $ >
fi
V:,&2V;T-3V-&2$+
J;“,
(35)
d2ffk dr2
---in 1 7i2 pE
+pkLwY
=2
h2 (
V$’ - 4ViT - 2V;; + 4ViT + 6V;& + 4;;
p: [
(
>
I)
+ vz!L!d
+ ,$
s:
2V-j--4V;T-6V:s:-4;f )
r,“,
(36)
in the triplet channel, with (k, 7’) = ( + , 0 ) and ( - ,1) ,,lf being the Lagrange multipliers. In these last expressions we have obtained an hermitian form for the differential equations
M.C. Bomi /Realistic interactions
334
TABLE 1 Energy per nucleon calculations, in MeV, for Urbana and Argonne 0~4interactions. The center-of-mass correction has been considered 014
Eo
E2p211
d(fm)
/iw(MeV)
Urbana Argonne
-4.58 -4.35
-5.32 -5.08
2.17 2.28
23 22
by using the set of modified
correlation
functions,
Jk (~1 = C&(r) )112J;k (rf .
(37)
At this stage the numerical calculation can be performed, as described in I. In this case we have imposed the sequential condition (again see I and references therein)
t3(ijlFi2’(1,2)
- ljij-ji)
= 0,
(38)
in order to have control over the cluster expansion, which we have truncated at the second order. Note that this condition allows us to select the particular value of the healing distance d which satisfies the constraint for each value of the variational parameter hw and, in this way, we can look for the minimum of the energy. The energies obtained at this first level of the calculation will be referred to as Ea. Once all the variational parameters and correlation functions have been determined, we correct the result for the ground-state energy by allowing the mixing of the ~0 state with the V2p2hstates. The energies calculated in this way will be denoted as Ezp2h. Finally, we obtain the excitation spectra by considering the whole basis allowed in each case: vlpih states for negative-parity
excited states and ~0 and v&,zh states for positive parity.
3. Results and final comments Fig. 1 shows the energy per nucleon EO as a function of the parameter fro for both the Urbana (solid curve) and the Argonne (dashed curve) ~14 interactions. In Table 1 we show, for each interaction, the minima of EO and E 2$h together with the co~esponding values of the fiw and d parameters. The results do not include Coulomb terms, and the center-of-mass correction is taken into account by simply substracting $ho, since we have chosen the single-particle wave functions as harmonic-oscillator states. As is apparent, our results are considerably far from the experimental value (- 7.07 MeV/nucleon). The reason for this discrepancy is not fully understood, however some points deserve commenting upon. Firstly, we have not estimated the importance of many-body cluster terms. Though we have tried to minimize their effects through the sequential condition, Eq. (38), sundry calculations have shown that their consideration increases the binding energy, and the
335
MC. Romi 1 Realistic interactions -1.0
-2.0
F z i
-3.0
W
-4.0
-5.o_ 5.
10.
15.
20.
25.
pie
[MeV]
30.
35.
40.
Fig. 1. The Eo energy per nucleon as a function of the variational parameter fro; the center-of-mass correction has been considered; full (dashed) line corresponds to Urbana (Argonne) v14interaction.
same occurs if a three-nucleon potential is added to the two-nucleon one [10-l 31. As an example, here we mention a recent work of Pieper et al. [ 141 in which the values E/A = -5.8 MeV for I60 and E/A = -5.6 MeV for 4He have been found with the Argonne 014 interaction. In this calculation the central correlation and exchanges are treated to all orders by Monte Carlo integration state-dependent of the Urbana
correlations,
VII three-nucleon
by 1.2 MeV/nucleon
and a cluster expansion
which include an L-S term. Furthermore, potential
[ 15 J is added, the binding
for I60 and 1.4 MeV/nucleon
for 4He. Further
is used for the
if the contribution energy increases calculations
with
more accurate forms of the wave function show that three-body correlations produce a signi~cant increase in the binding energy [ 161. Clearly, any calculation based on clusters expansions must guarantee the convergence of the cluster series. Secondly, our two-body factor correlation in Eq. (7) should have a full 014 structure to be consistent with the interaction, or, at least, L 1S pieces, which will produce new kinds of terms in the effective potential. The absence of these components could be one of the reasons for the lack of binding energy shown by the 014 interactions with respect to our previous calculations with 0s interactions. Although we have not tested how these new terms affect the energy, we expect the effects to be small, but not negligible. The Ezp2h results in Table 1 show, as for the 08 interaction, that the correct construction of the nuclear wave function plays an important role. In this respect the admixtures of
M.C. Boscci /Realistic interactions
336
TABLE 2 Kinetic energy and cont~butions in MeV/nucieon of each component of the ut4 interaction to the Eo value of the energy
014
hw(MeV)
Urbana Argonne
23 22
Contributions
T 14.18 18.51
VC
-35.72 -24.08
VLS
VT
-8.77 -23.91
-0.10 0.56
VL
1.21 2.99
VLS2
-0.18 -2.23
cv
-43.56 -46.47
TABLE 3 in MeV/nucleon of each component of the effective potential V:& to the Eo value
v14
fiw(MeV)
Urbana Argonne
23 22
-30.66 -29.74
0.26 0.02
1.02 1.66
-29.38 -28.06
-4.58 -4.35
TABLE 4 Same as Table 1 for Iio = 15MeV Eo
E2,,2h
-3.21 -3.16
-3.81 -3.76
014
Urbana Argonne
d(h)
2.05 2.17
states produce a non-negligible effect, the corresponding gain in energy being about 0.7 MeV per nucleon; this fact can be related to the extreme simplicity of the correlation operator used. Tables 2 and 3 show the contribution in MeV/nucleon of each component of the u14 interaction and V$, potential, respectively, to the E. value of the energy. These contributions are related by E. = %co 64
=
+ Ef
c v, , eff
(401
K
where K = {C, T, LS, L, LS2). From these tables we can conclude that, as expected, the central and tensor parts of the interaction give the most important contribution to the total energy, the contributions of the remaining terms being non-negligible.
337
;I= I--
2-_ o-_ 1--
2-4-----
z-4--
2-4--
3-_ 2-
3-_
3-
2-
2--
0-_ ‘2--
o- = 2-
1------
1--
3-----
3--
2p’----
-
20-1-3--
9
F
I-2--
t
1-----
6
3--
~
3
Exp.
I’rb-VI4
ArgV1.l
RX-V8
-
Fig. 2. Low negative-parity states with isospin T=O.
The optimal
values found for frw in the ground-state
minimization
are too large (al-
though fiw does not correspond to the mean-field ha~oni~-oscilIator potential, because of the central-field modification introduced by the Jastrow correlation through the CBF mechanism). This fact may be due to the inadequacy of harmonic-oscillator single-particle states, and a way to improve this deficiency would be to carry out a Hartree-Fock determination of single-particle
wave functions
from the Jastrow generated effective interaction,
which will imply a highly non-trivial double self-consistent calculation, given that the effective interaction depends on the correfation function, which in turn depends on the nuclear density. Few calculations of energies of light nuclei using 014 interactions exist to compare our work with, although most of the theoretical effort has been devoted in recent years [ 1O161 to the aIpha particle. The corresponding variationai results for the ground-state energy are of 5.7 [ 151 and 5.7 [ 151-5.8 [ 131 ~ev~nuc~eon with the Urbana and Argonne at4 interactions, respectively. These values are lowered to 7.3 [ 151 and 7.1 [ 15]-7,O [ 131
M.C. Bosch / Realistic interactions
338
E
I
i
[M&l -
1
T=l
1-_
21 -
;‘4-
21 -
‘L-. 3--_
18 -
,-,2-_
2_
“-
1-
1-1-
2--
7-p
9--
73-
4-‘A3-----~~
1I--
15 -
? :j--
12 -
9--
‘-
o-
I
Fig. 3. Low negative-parity states with isospin T= 1.
MeV/nucleon, respectively, when the three-nucleon Urbana VII potential is added. Our calculations for the alpha particle and other nuclei in the p-shell [ 171 also show a lack of binding energy in comparison with those results previously mentioned, making it clear that our choice of the correlation factor is inadequate. The calculated spectra are shown in Figs. 2-4, together with the experimental
levels
[ 181 and the previous us calculations [ 51. A value of hw = I5 MeV has been used for both interactions considered. The reason for choosing this value instead of our optimal values shown in Table 1 is twofold. Firstly, the levels theoretically determined by using the optimal values are too high in excitation and, secondly, this was the value used in II for the comparison with several other ug calculations. Table 4 shows the corresponding energy values for this choice of fiw. As is apparent from these figures, the new terms in the potential have not been able to improve the results previously obtained for the 0s interaction, at least at the level of the calculation described above. As in the us interaction, the present approach fails completely in the quantitative description of the excitation energies, and a large shift of the whole calculated spectrum is found with respect to the experimental data. The reasons for this shift, which is unlikely to be resolved by adding components to the potential or to the correlation operator, are not clear. It would probably be convenient to consider other effects such as the excitations of the center-of-mass motion and, in addition, to try
339
1+-
1*-
2*I+-
2+---I”= ,g+--...--
4+2f
2+
3+-
:3+-
0+-
0+-
2+-
.>+_
p----0*5
t
1
&-VI
Esp.
1
Fig. 4. Low positive-parity states with isospin T=O.
different formuIations of the CBF scheme. In any case, and before tackling all the consequent work involved, we are interested in testing if the new terms in the potential, at the level of calculation that we have exposed here, have a significant repercussion on the co~espo~di~g excited spectra of the other nuclei in the p-shell with respect to the results previously obtained for the us nucleonnucleon potentials. Work in this direction is in progress. The author is grateful to R. Guardiola for his very valuable guidance, and also wishes to thank A.M. Lallena for his assistance in the drawings. A critical reading of the paper by S. Fantoni is acknowledged.
References il] 1.W. Clark and P. Westhaus,
Phys. Rev.
141 619661 833; 143 tg966) %I
340
M.C. Bomi /Realistic interactions
[2] J.W. Clark in Progress in particle and nuclear physics, vol. 2, ed. D.H. Wilkinson (Pergamon, Oxford, 1979) [3] S. Fantoni, B.L. Friman and V.R. Pandharipande, [4] [5] [ 61 [7] [8] [9]
[lo] [ 111 [ 121 [13] [ 141 [ 151 [16] [ 171 [ 181
Nucl. Phys. A386 (1982) 1; A399 (1983)
t.C. Bosch and R. Guardiola, Nucl. Phys. A476 (1988) 471 R. Guardiola and M.C. Bosch, Nucl. Phys. A489 (1988) 45 I.E. Lagaris and V.R. Pandharipande, Nucl. Phys. A359 (1981) 331 R.B. Wiringa, R.A. Smith and T.L. Ainsworth, Phys. Rev. C29 (1984) 1207 J.W. Clark and P. Westhaus, J. Math. Phys. 9 (1968) 131,149 M.L. Ristig, W.J. Ter Low and J.W. Clark, Phys. Rev. C3 (1971) 1504 J. Carlson and V.R. Pandharipande, Nucl. Phys. A371 (1981) 301 J. Carlson, V.R. Pandharipande and R.B. Wiringa, Nucl. Phys. A401 (1983) 59 R.B. Wiringa, Nucl. Phys. A401 (1983) 86 J. Carlson, Phys. Rev. C38 (1988) 1879 SC. Pieper, R.B. Wiringa and V.R. Pandharipande, Phys. Rev. Lett. 64 (1990) 364 R. Schiavilla, V.R. Pandharipande and R.B. Wiringa, Nucl. Phys. A449 (1986) 219 J. Carlson, Nucl. Phys. A522 (1991) 185~ M.C. Bosch, to be published F. Ajzenberg-Selove, Nucl. Phys. A460 (1986) 1